Testing a specified difference between proportions

Testing of a specified difference between binomial proportions (including the test of equality) may be formulated to preserve the asymptotic power intended for pairs of simple hypotheses related under a specified family of alternatives, such as simple shift alternatives. This formulation simply matches parameterizations for estimated and true response rates in terms of corresponding specifications for the offsets realizing the specified difference from a nuisance parameter. The parameterization is then employed in estimating the variance of the difference in proportions. An unbiased estimator of the appropriate variance is provided and asymptotically unbiased hypothesis testing is verified.